Factor the following expression: $2$ $x^2$ $-7$ $x+$ $6$
Explanation: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(2)}{(6)} &=& 12 \\ {a} + {b} &=& & & {-7} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $12$ and add them together. The factors that add up to ${-7}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-3}$ and ${b}$ is ${-4}$ $ \begin{eqnarray} {ab} &=& ({-3})({-4}) &=& 12 \\ {a} + {b} &=& {-3} + {-4} &=& -7 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {2}x^2 {-3}x {-4}x +{6} $ Group the terms so that there is a common factor in each group: $ ({2}x^2 {-3}x) + ({-4}x +{6}) $ Factor out the common factors: $ x(2x - 3) - 2(2x - 3) $ Notice how $(2x - 3)$ has become a common factor. Factor this out to find the answer. $(2x - 3)(x - 2)$